AD-AO92 596 TECHNION - ISRAEL INST OF TECH HAIFA DEPT OF AERONAU-ETC F/0 12/1THREE DIMENSIONAL OPTIMAL PURSUIT AND EVASION WITH BOUNDED CONT--ETCtU)NOV 79 .J SHINAR, S GUTMAN F99620?79-C0135

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Missile Avoidance, Control Theory, 3-D Equations, Linear Kinematic, BoundedLimits, Stochastically Solution

j0. ABSTRACT (Continue on reverse side If necessary end identify by block number)

'The missile-aircraft pursuit-evasion problem is formulated by a three-dimen-sional linearized kinematic model with bounded control. The formulation isvalid both for the optimal control (against a known adversary strategy) and thezero sum differential game versions. Assuming perfect information the linear-ized kinematic model yields for both versions a solution which can be imple-mented in real-time for airborne application. The avoidance of a known pursuerby an evader who has no state information is solved by a stochastically optimalperiodical maneuver. Other examples of imperfect information are briefl yONT)

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THREE DIMENSIONAL OPTIMAL PURSUIT AND EVASION

WITH BOUNDED CONTROLS*

J. Shinar** and S. Gutman***

Technion -Israel Institute of Technology

Haifa., Israel

T.A.E. Report No. 381

*Research partially sponsored by AFSC Contract No. F49620-79-C-013S

**Associate Professor, Department of Aeronautical Engineering

**Senior Lecturer, Faculty of Mechanical Engineering.

1,

ABSTRACT

The missile-aircraft pursuit-evasion problem is formulated by a three-

dimensional linearized kinematic model with bounded control. The formulation

is valid both for the optimal control (against a known adversary strategy)

and the zero sum differential game versions. Assuming perfect information

the linearized kinematic model yields for both versions a solution which

can be implemented in real-time for airborne application. The avoidance of

a known pursuer by an evader who has no state information is solved by a

stochastically optimal periodical maneuver. Other examples of imperfect

information are briefly discussed.

or~jo

TX edT

CC~a

S E-ECTED

-DE 5 IM

TABLE OF CONTENTS

ABSTRACT .. ...................... ........ i

TABLE OF CONTENTS. .. ................... .. .. . ...

1. INTRODUCTION. ..... ................ .. .. .....

2. MATHEMATICAL ANALYSIS. .. ...................... 4

A. Two-Person Zero-Sum Linear Differential Gamnes. .. ........ 4

B. Linear Optimal Control .. ................... 6

3. APPLICATION TO 3-D PURSUIT-EVASION .. ............... 8

A. Differential Game Version. .. ................. 8

B. Optimal Control Version .. ..... ............. 13

4. DISCUSSION OF THE DETERMINISTIC RESULTS .. .... ........ 14

A. interpretation. ... ..................... 14

B. Implementation. ..... ................... 16

5. MISSILE AVOIDANCE WITH IMPERFECT INFORMATION. ... ....... 17

6. CONCLUDING REMARKS. .. ...................... 18

REFERENCES. ... ........................... 20

LIST OF FIGURES .. .. ......................... 23

am-.

1. INTRODUCTION

The research effort to solve the missile-aircraft pursuit-evasion problem

has been directed towards two important applications: (a) optimization of the

control law for guided missiles; (b) enhancement of aircraft survivability

by missile avoidance. The problem can be formulated either as a one-sided

optimization, i.e., optimal pursuit1, 2 and optimal evasion,3- 6 or as a zero-

sum differential game. 7-11 Regardless of the formulation the problem is

inherently complex. Thus the solution has to be attempted in 3 consecutive

phases:

1. Modelling, which preserves the essential features of the original problem;

2. Analytical solution of the simplified, but representative mathematical

model;

3. Implementation of the results derived from the analysis.

The objective of this paper is to discuss all 3 phases of the solution

and to present some recently obtained results.

The missile-aircraft engagement is characterized - unlike other pursuit-

evasion problems (as the "homicidal chauffer," the game of two cars, 12 etc.) -

by a "fast" pursuer (VP/VE > 1), which also has a higher lateral acceleration

capability (ap/aE > 1). Analyses based on pure kinematic models (neglecting

vehicle dynamic response) have demonstrated13' 14 that,due to such superiority

of the pursuer,point capture of the evader can be guaranteed. Yet whenever

pursuer dynamics has been taken into account , it has been shown that a

non-zero miss distance can be achieved by appropriate evasive maneuvering.

r2. . . . .. , 'i. -. .4- - -

• • ° "

-2-

Moreover, a recent study6 clearly indicated that optimal missile avoidance

requires a three-dimensional strategy.

Thus, two attractive and frequently used simplifying assumptions as:

(i) neglecting pursuer dynamics, 7- 9 13, 14 (ii) two-dimensional analy-

sis"-" 9,10,13 are not made in the mathematical model used in this paper.

Fortunately, the very nature of the missile-aircraft engagement does jus-

tify the trajectory linearization used in many studies.1-4'611

As a consequence of these major modelling considerations the missile-

aircraft pursuit-evasion is formulated and solved in the following sections

as a 3-D, linear differential game. These sections also include brief com-

parisons with the one-sided optimal control formulation.

The assumptions used in the analysis can be summarized as:

1. Both pursuer and evader are considered as point-mass vehicles.

2. The speed of each vehicle is constant, the pursuer being the faster

(VP /VE > 1).

3. The relative motion is three-dimensional (See Fig. 1).

4. Gravity, having no effect on the relative trajectory, is neglected.

5. The initial conditions of the pursuit are near to a collision course

F (See Fig.2).

6. Relative trajectory can be linearized around the initial line of sight

vector.

7. The lateral acceleration commands of both vehicles are bounded with

circular vectograms perpendicular to the respective velocity vectors

a a,,a > 1).

-3-

8. The pursuer's response to its acceleration command is approximated by

single time constant Tp.

9. Evader dynamics is neglected for sake of simplicity. (It has been

shown'4, 6 that evader dynamics has only a non-qualitative secondary

effect on missile avoidance).

10. The performance index of the problem is the miss distance (distance

of closest approach).

11. There exists perfect (complete and instantaneous information on the

state variables and the parameters of the problem. This assumption

is mandatory to obtain a solution in the deterministic sense.

Assumptions 7 and 8 present a new formulation of the three-dimensional

pursuit-evasion game compared to previous linear quadratic mathematical

models neglecting pursuer dynamics.7'8 The validity of the other assump-

tions is discussed in detail in previous papers dealing with optimal eva-

sion control.4'6

The generalized analytical solution for this class of problems, which

has been developed in recent papers, 1 ' 1 s is briefly summarized in Section

2 and applied to the three-dimensional pursuit evasion problem of interest,

defined by the above listed assumptions, in Section 3.

In Section 4 the characteristics of this entirely new solution of the

deterministic linearized differential game are discussed and in the sequel

the implementation of the optimal strategies is considered.

Since implementation depends on the available information, the difficul-

Pties created by non-complete information are examined. In Section 5 an

example of a recent stochastically optimal solution is presented for mis-

sile avoidance without state information. Other cases of imperfect in-

formation are also discussed briefly.

.................... .....

-4-

2. MATHEMATICAL ANALYSIS

A. Two-Person Zero-Sum Linear Differential Games. i

It is assumed that the reader is familiar with basic results in two-

person zero-sum differential games.'12 16 Let a game of fixed duration be

defined by

x=A(t)x + B(t)u +C(t)v, x(O) =xO

u EU , v EV1()

J = IIDx(T)II

where matrices A, B, C, D, have proper dimensions and A(-), B(-), Q() are

continuous.I

It is required to find among all admissible strategies {p(-), e(-)},

such that u(t) = p(x(t),t), v(t) = e(x(t),t), an optimal strategy pair

{p*(.), e*(*)} satisfying the saddle point inequality

J(x,t,p*,e) S J(x,t,p*,e*) =J*(x,t) .j J(x,t,p,e*) (2)

In order to simplify (1), let

y = D (T,t)xI

B(T,t) = Dci(T,t)B (3)

C(T,t) = D4'(T,t)C

where 4'(T,t) satisfies

dt

S -

-5-

By (3) the game (1) is transformed to

= B(T,t)u + C(T,t)v , y(O) = Yo

u E U , vEV E (5)

J = IIy(T)R

For this formulation a candidate optimal strategy pair is given" by

Min E*'B(T,t)u = C*'B(Tt)p*uEU

(6)

Max *'C(T,t)v = F*C(T,t)e*

where both the vector F* and the value of the game J* are determined by

J* = Sup V'y+ f [Min ' B(T,T)u + Max E' C(T,T)v dT} (7)

RE11=l t uEU vEV

Now a sufficiency theorem on the existence of the saddle point solution is

presented.

Definition: A tube IT is defined by

n = {(y,t) J*(y,t) = c,c > 0

T . = {(y,t) : J*(y,t) < c,c > 0

iTo = {(y,t) J*(y,t) > c,c > 0

Based on this definition, the following theorem has been proved. 1

Theorem 1. Consider game (5). Let y(t) E n and suppose that iT is C1. Then

-6-

(M) {p*('),e*(.)} given by (6)-(7) is an optimal strategy pair;

(ii) p*(') guarantees the minimizer a cost not higher than c, while

e*(.) guarantees the maximizer a cost not lower than c;

(iii) Yo E i. implies that a solution y('), generated by {p*,e} remains1

in 7i; likewise, Yo E 7o implies that a solution y(-) generated

by {p,e*} remains in io.

A direct consequence of this theorem is that along an optimal trajec-

tory, E* is a constant unit vector.

Remark 1. In case 7 is only piece-wise smooth, Eq.(7) still determines the

optimal cost J* and the vector E* at every point along those optimal tra-

jectories which do not intersect the region of non-smoothness. (For a more

detailed discussion on this subject, see Refs. 16 and 17).

B. Linear Optimal Control

If one of the participants has a fixed strategy known by his adversary

the problem reduces to a one sided optimization. In this subsection the

optimal control problem is formulated from the maximizer's viewpoint (a

similar formulation for the minimizer will not be repeated). Assuming

that the minimizer has a known linear feedback control law u= K(t)x (with-

out hard clnstraints being imposed), the one sided version of (1) becomes

x = A(t)x + C(t)v x(o) =xo

v E Vf (8)

J = IIDx(T) H

with A(t) = A(t) + B(t)K(t).

-7-

A transformation similar to (3) simplifies (8) to

y = C(T,t)v y(O) = yo

v EV 1(9)J = y(T)h

For this formulation the optimal control v*(t) is given by

Max &'T(T,t)v *C(T,t)v* (10)vEv

where * and the optimal cost J* are determined by

J*(y,t) = Sup lIy + Max &' C(T, T)vj dT) (11)tj LvEV

Along an optimal solution &* is a constant unit vector. It is important to

note that the one sided optimization is valid (unlike the differential game

version) without any further assumption on the smoothness of J(-).

-8-

3. APPLICATION TO 3-D PURSUIT-EVASION

A. Differential Game Version

In this sub-section the missile-aircraft pursuit-evasion is solved based

on the set of assumptions 1-11 as a zero-sum perfect information linear dif-

ferential game in three-dimensional space (Fig. 1). By Ass. 5 the analysis

is restricted to the neighborhood of collision course (Fig. 2) where the

relative trajectory can be linearized (Ass. 6). As a consequence of the

linearization, the relative motion in the line of sight direction (the X

axis) is of constant speed and the duration of the game T is determined. The

dynamics to be considered is perpendicular to the line of sight. In this

coordinate system the control vectograms, which are by Ass. 7 circular, per-

pendicular to the respective velocity vectors, become elliptic (See Fig. 3).

Accordingly, the equations of motion of the three-dimensional pursuit-

evasion have the form of (1) with

U u : u'Ru < a2 }

(12)

V- {v : v'Sv <a2 }

where the matrices R, S are symmetric and of proper dimensions. By defining

(see Fig. 1)

y _ y _ Y E Z _ Zp ZE (13)

the state vector x is of 6 components

x= [,,Yp" ZZZ (14)' P1

-9-

The matrices in (1), (4), and (12) are

with

0 1 0Al 0 0 1 FD 0 0 0 00i~ 00 1 0=

0 lIT 1 0 0 : 0

BI I/T[ : i: 0

[0 0 :0 0 0 ]

0 0 0 0 -1

R=[1/COS2XP(o) 0] S l/Cosx2EX ) 0]

Note that V P/V E> 1 (Ass. 2) implies that cosX P(0) > coSxE (0) (See Fig. 2).

Equations (3) have the form

= = Y + (T::)Y + :13Y (15

B=(0 1 3/T ,)1 2 ; 0 12 12 (16)

wih012 T 013 T 2 1P(0) (17)

p p

10-

where

6 T-t

p (18)

is the normalized "time-to-go" and

A -O

4)(O) _ 0 + e 1 > 0 V 0 > 0 (19)

Let R-1 = MI M and S- ME M

[ XP ] ; ME= [osXE(o) 0]

Then equation (6) is satisfied (using 8 rather than t) by

p*(y,O) = - aRMl'p IM p *I

(20)

e*(y,O) =- a M.' MEE E M *ff

E

where t* and the value of the game are determined by

J*(yO) Sup { 'y - a(O) Mp , B(O)IM EI} (21)

with

c(0) a T2 2(n)dn -a -2,= apTp J ipr2r at~[

e (22)S(O) a, 2 202ndn aETp 2aETp 0

I, --- J

- 1 1 I

It can be directly concluded from (21), (22) that at y = 0, *' =

= (0, ±1); i.e., the optimal control vectors are perpendicular to the no-

minal collision plane. Moreover, the same conclusion holds for {y: Yi = 0}.

The existence of such a saddle point solution can be proved (based on

the definition of the tube n) via Theorem 1 of the previous section, if all

tubes are smooth (see also Remark 1).

For this particular problem a simple test on (21) shows that at some

subregions of {Y;Y2 = 0} (containing y = 0), E* is not unique; hence

tubes n in these regions are not smooth. Since the investigation of piece-

wise smooth tubes is not the aim of the present paper (Refs. 16 and 17 deal

with this topic), we slightly modify the control set U, V, in order to avoid

this phenomenon. In a usual situation ap cos Xp(O) > aE (see Fig. 3).

Based on this observation, we reformulate the problem from the pursuer's

point of view. Replacing in (12) the matrices R and S by the identity ma-

trix yields circular vectograms and consequently,

MP = ME = 12 (23)

If we further replace in (12) ap by

p P, ap cos Xp(0, (24)

the "worst case" version of the original problem is obtained from the

pursuer's point of view (aE = aE). For this formulation

y ^(25)lyl

and consequently the optimal strategies and the Value of the game are given by

~[

- 12 -

p (y,0) = - py

(26)

*(y,e) = - -

J*(y,B) = ly ae) + B(o) (27)

with 3p replacing ap in (22).

Analysis of the tubes shows that there exists some cm such that a tube

11(C = cm) is tangent to {(y,6) : y = 0} at 8 = 8 . Such tube is called the

"minimal tube.""1 The point of tangency 8 = 8 can be easily computed bys5

setting in (27) J* = cm

lyl = c + a(e) - 8() (28)

8 is now defined by

W = 0 (29)

yielding

aE-06

~P

The minimal cost c is obtained by substituting (30) and y = 0 into~m

Eq.(28)

Cm ae s +[ i(- (31)= Ep , 1 2 - E

The dependence of e and c ,fc/aET2 on the maneuver ratio (aE/aP) iss m E P

depicted in Fig. 4.

4 l ~ ~-- . . .. S . . ,

- 13 -

It is easy to see from (27) that all tubes 7r (c > cm) are smooth. As

a consequence of the existence of the "minimal tube" the (y,O) state space

can be decomposed into D and D , being defined as

Dc {(y,e)ir :c. > c }U{(ye) : 0 < s ) (32)

The optimal strategy pair j*(.), *() is given by (26) in Dc and by any

arbitrary admissible pair in D. The Value of the game J (i.e., the miss

distance guaranteed to players using the optimal strategy pair) is de-

termined by (27) in Dc and by cm = const. (31) for every point in D.

B. Optimal Control Version

In the investigation of the avoidance of a proportionally guided mis-

sile, one deals with a model of the form given in (8). Following the

notation of the previous sub-section, the solution has the form

ME *

e*(y,O) - aEE IM E sgn i12 (0) (33)

where the constant vector * and the optimal cost T* are given by

j*(y,6) = Sup {E'y + "()IME&I1 (34)I11

6

= aETP IT , (n) Idn (35)

As in the game version, it is clear that at y = 0, ' u (0,1), i.e.,

the optimal control is directed perpendicular to the initial collision

hLim

14-

plane. Eq. (34) is an extension of the solution obtained in Ref. 6, which

was derived for yo 0.

It is very important to observe that i12 (0) is not the same as 012 in

(17) and as a consequence it is no longer positive for all 0. As a matter

of fact 012(0) is a switching function (computed in Ref. 4).

4. DISCUSSION OF THE DETERMINISTIC RESULTS

A. Interpretation

The deterministic solution of the differential game version gives the

optimal strategies and the resulting miss distance as a function of the

"reduced state" y and the normalized "time-to-go" e. The vector y can be

interpreted as the "predicted miss distance," being composed (see Eq. (15))

of the "zero-effort miss" and an acceleration term compensating for pursuer

dynamics.

The decomposition of the (y,O) state space is of major significance.

The domain V is characterized by "long" pursuit times 0 > 0s and small or

moderate deviations from collision course. This region (where according

to Ass. 5 most trajectories begin) is dominated by the pursuer, who can

reduce, for any trajectory initiated in D, the predicted miss distance to

zero at 0 = 0s against any admissible evasive maneuver. Though the pursuer

strategy can also be an arbitrary admissible one in V, an adaptation of a

linear time varying feedback control law as proposed for the 2-D casel'

may be an attractive choice. This strategy has some similarity with the

guidance laws obtained by one sided linear quadratic optimization.1'1x

~~;

i - 15 -

As every trajectory initiated in D passes through the point (y = 0,

0 = s), they all have the same strategy for 0 -< 6 s and consequently an

outcome independent of the initial conditions. In the original formula-

tion (Eqs. (12)-(22)), the terminal strategy is given by E = 0, i.e.:

hard maneuvering perpendicular to the initial collision plane. This veri-

fies the recently obtained result for optimal missile avoidance,6 and con-

firms the necessity of 3-D analysis. The direction chosen is the one where

the evader-pursuer maneuver ratio is maximal (see Fig. 3), being the conse-

quence of (21) for y = 0.

The main limitation of the present analysis lays in the trajectory lin-

earization (see Ass. 6). In a previous paper, 6 dealing with the optimal

control version of the same problem, it was concluded that trajectory lineari-

ztion is justified if two conditions are satisfied:

1. The dynamic similarity parameter of the problem,1' which can be inter-

preted as the maximum direction change of the evader during the period of the

pursuer's time constant, is small.

2. The optimal solution does not predict excessively long maneuvers in

any specific direction.

The first condition can be observed before the linearized trajectory as-

sumption is adopted. The second one, however, requires an "a posteriori"

verification. The total change of interception geometry has to be computed

and examined to decide upon the validity of the linearization.

In order to put the present analysis in the proper context with some

"classical" missile avoidance concepts the following remark seems to be of

relevance. V,. ) superiority of the pursuer (VP > VE) and the constant

speed hypothesis (See Ass. 2),used in the mathematical model, clearly indicateI

16-

that the initial conditions of the engagement has been chosen well inside

the effective firing envelop of the missile, ruling out the so called

fleeing tactics of the evader. Such tactics, having the objectve to

convert a head-on engagement into a tail-chase, hope to profit from

eventual missile slow down. They can be effective only against mis-

sile systems which use an "ill determined" (constant target direction)

firing zone concept.

B. Implementation

Optimal strategies are based on perfect knowledge of the reduced state

variables and the "time-to-go". The computation of "time-to-go" requires

range and range-rate measurements. The "zero-effort miss" can be computed

if the rate of turn of the line of sight is also measured. These measure-

ments, as well as of the actual missile acceleration, are of common

practice in many guidance systems. Thus the implementation of the optimal

pursuer strategy given by (25) does not seem to present difficulties. The

efficiency of such scheme will however depend on the accuracy of the meas-

urements or the estimation process.

By contrast, the implementation of the corresponding evasion strategy

is a rather formidable, if not impossible, task. Although some recent ef-

forts have been made to estimate the "zero-effort miss" and the "time-to-

go" onboard of aircraft by appropriate filtering,12 the estimation of mis-

sile acceleration is beyond the state of art. Moreover, the optimal strat-

egy assumes perfect knowledge of the missile parameters. It has to be based

on previous intelligence as well as on real-time identification of the

N

17 -

particular type of missile to be avoided. If intelligence is available,

it may also include data on the actual guidance law, reducing the problem

to a one sided optimization,3-6 where the evader can achieve (assuming

perfect information) larger miss distances than guaranteed by (26) and (27).

It can be concluded that for missile avoidance applications the hypo-

thesis of perfect information is a most critical one and in many cases it

cannot be justified. Therefore, the problem has to be re-examined with

non complete information.

I:

5. MISSILE AVOIDANCE WITH IMPERFECT INFORMATION

In the optimal missile avoidance problem there are several sources of

information imperfections:

a) Lack of intelligence or identification 'parameter uncertainty);

b) Unaccessible state variables (partial observability);

c) Unaccurate state variables (measurement errors or noise);

d) Non existent threat warning (no initial conditions).

Each of these topics deserves a separate analysis and a detailed dis-

cussion. In the limited scope of this paper it seems, however more produc-

tive to present an example which illustrates the relative importance of

perfect state information to successfull missile avoidance.

For this purpose it is assumed that the evading airplane has no infor-

mtion on the relative state (including initial conditions), but the pursuer

parameters are known. This problem has been recently solved" by determin-

ing a periodical random phase maneuver which maximizes the R M.S. value of )A' I

r .:

-18-

the miss distance. It has been shown that such periodical strategy is

much superior to the Random Telegraph Maneuver mentioned in previous

works20 21 and it is indeed of optimal structure. The optimal frequency

w* and the resulting miss distance distribution can be computed as func-

tions of pursuer and evader parameters.

Comparing the results of such stochastic optimization to the case of

perfect information (for an example of two-dimensional missile guided by pro-

portional navigation, N' = 4.0) reveals that the R.M.S. miss distances rech

60-80% of the optimal deterministic value. This comparison indicates that

the degradation of missile avoidance capability due to imperfect informa-

tion may not be as serious as it could have been estimated. If the avoid-

ance efficiency could be partially retained in the stochastic sense even

for a total absence of state information, there is a definite hope that bet-

ter (and probably satisfactory) performance could be achieved if partial or

noise corrupted state measurements do exist.

6. CONCLUDING REMARKS

In this paper the 3-D missile-aircraft pursuit-evasion is solved in a

closed form as a linear zero-sum perfect information differential game. It

is the first time that such solution of this problem with a realistic math-

ematical model, taking into account hard control constraints and pursuer

dynamics, is presented. The compact closed form solution is an application

of a recently developed methodology,1 ' 15 which can be equally used for the

one sided optimal control formulation of the problem. The solution of the

- 4 - -K* I *r-

19 -

game has a very clear geometrical interpretation creating a new insight

for the missile-aircraft pursuit-evasion. As a consequence, the applicabi-

lity of the results can be directly discussed.

The optimal strategy for the pursuer, proposed by the closed form solu-

tionseems very attractive for guided missile applications. On the contra-

ry, the implementation of the deterministic optimal missile avoidance strat-

egy presents serious difficulties due to incomplete available information.

In the paper the sources of imperfect information are discussed and for

the "worst case" example of total absence of state information a stochasti-

cally optimal solution is given. Comparison to relevant deterministic

results shows that the degradation of avoidance efficiency, due to lack of

information, is not overwhelming. It implies a potential of better results

for the case of partial or noisy state measurements and parameter uncer-

tainties. These problems require an extensive future research.

5,,-*.... . 1

-r ... ..-. - .....--

200

- 20 -

REFERENCES

1. Cotrell, R.G., "Optimal Intercept Guidance for Short Range Tactical Mis-

siles," AIAA Journal, Vol. 9, No. 7, July 1971, pp. 1414-1415.

2. Asher, R.B., and Matuszewski, J.P., "Optimal Guidance with Maneuvering

Targets," Journal of Spacecraft and Rockets, Vol. 11, No.3, March 1974,

pp. 204-206.

3. Slater, G.L., and Wells, W.R., "Optimal Evasive Tactics against a Pro-

portional Missile with Time Delay," J. Spacecraft and Rockets, 10, 5,

May 1973, pp. 309-312.

4. Shinar, J., and Steinberg, D., "Analysis of Optimal Evasive Maneuvers

Based on a Linearized Two-Dimensional Model," J. Aircraft, 14, 8, Aug.

1977, pp. 795-802.

5. Steinberg, A., Forte, I. and Shinar, J., "A Qualitative Analysis of

Optimal Evasion in 2-D," 20th Isr. Ann. Conf. on Aviation and Astro-

nautics, 22-23 Feb., 1978, pp. 111.

6. Shinar, J., Rotsztein, Y., and Bezner, Z., "Analysis of Three-Dimen-

sional Optimal Evasion with Linearized Kinematics," J. Guidance and

Control, Vol. 2, Nov-Dec. 1979.

7. Ho, Y.C., Bryson, A.E., and Baron, S., "Differential Games and Optimal

Pursuit-Evasion Strategies," IEEE Trans. of Automatic Control, Vol.

AC-10, 4, Oct. 1965, pp. 385-389.

8. Aumasson, C., "Approche du Problem de l'Interception Tridimensionnelle

Optimale par la Th~orie des Jeux Differentiels Lineares Quadratiques,"

La Recherche Aerospatiale, No. 1976-6, Sept.-Oct., pp.255-266.

- 21

9. Gutman, S., and Leitman, G., "Optimal Strategies in the Neighborhood of

a Collision Course," AIAA J., 14, 9, Sept., 1976, pp. 1210-1212.

10. Gutman, S., "On Optimal Guidance for Homing Missiles," J. Guidance and

Control, Vol.2, July-Aug. 1979, pp. 290-300.

11. Gutman, S., "Remarks on Capture-Avoidance Games," (to appear in

J.O.T.A.).

12. Isaacs, R., Differential Games, Wiley and Sons, New York, 1965.

13. Cocayne, E., "Plane Pursuit with Curvature Constraints," SIAM J. Appl.

Math., Vol.15, No.6, Nov. 1967, pp. 1511-1516.

14. Rublein, G.T., "On Pursuit with Curvature Constraints," SAM J. Con-

trol, Vol.10, No.1, Feb. 1972, pp. 37-39.

15. Blaquiere, A., Gerard, F., and Letman, G., Quantitative and Qualitative

Games, Academic Press, New York, 1969.

16. Gutman, S., "Linear Differential Games with Bounded Controls," TME Re-

port No. 365, Faculty of Mechanical Engineering, Technion, Israel Inst.

of Technology, Haifa, 1979.

17. Shinar, J., Gutman, S., and Medinah, M., "On Singular Surfaces in a

Linear Differential Game with Asymmetrically Bounded Control," TAE Re-

port No.387, Department of Aeronautical Engineering, Technion, Israel

Inst. of Technology, Haifa, 1979.

18. Shinar, J., and Rotsztein, Y., "Similarity Parameters in Pursuit-Eva-

sion Problems," TAE Report No. 336, Department of Aeronautical Eng.,

Technion, Israel Inst. of Technology, Haifa, Apr. 1978.

19. Brown, C.M., et al, "Trajectory Estimation for Air-to-Air Threat/Col-

lision Assessment," Proceeding8 of NAE Conf. 77, pp. 340.

U-. . . .t

- 22

20, Bezner, E., "Optimal Evasive Maneuvers in Conditions of Uncertainty,"

M.Sc. Thesis, Technion, Israel Inst. of Technology, Aug. 1978.

21. Speyer, J.L., "An Adaptive Terminal Guidance Scheme Based on an Ex-

ponential Cost Criterion with Application to Homing Missile Guidance,"

IEEE Trans. on A-E., Vol. AE-21, No.3, June 1976, pp. 371-375.

22. Fitzgerald, R.J., and Zarchan, R., "Shaping Filters for Randomly

Initiated Target Maneuvers," AIAA Guidance and ControZ Conference,

Palo Alto, California, August 1978.

77,

-23-

LIST OF FIGURES

Page

FIGURE 1. 3-D Pursuit-Evasion Geometry. 24

FIGURE 2. Collision Course Geometry . . . .. . . 25

FIGURE 3. Acceleration Vectograns Perpendicular to the Initial Line of

Sight......_............... ........ 26

FIGURE 4. Dependence of e s and the Normalized Minimal Cost E on the Evader-

Pursuer Maneuver Ratio.. . . . ... . . . 27

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